The dominant binding site in cubic is shallow with respect to rotations in the face of the cube. Arrows are used to indicate transfers and rotations to equivalent binding sites. The large dark ions are oxygen ions. The smaller corner ions are the B-type ions. The lone ion in the center is an A-type ion. Hydrogen is shown in white.
Proton binding sites and transitions states in are shown on an unfolded perovskite cubic unit. The simulation box is comprised of four of these units placed together as described in Ref. 3. The proton binding site energy is shown on the bound oxygen. The actual geometry of the OH bond in the proton binding site is similar to that shown in Fig. 1 for the cubic case. Transition state energies are shown between oxygens. The transition state energies in the center of the cube are for interoctahedral proton transfer. Rotational transition state energies from one face to an adjacent face are written on the cube edges. All energies are in eV. To aid comparison with other papers, an arrow highlights the row of O1 sites.
The adjacency matrix formalism identifies one four-step path and three six-step paths that lead to conduction through the whole cubic perovskite. The four-step path, , is shown in (a). The six-step paths are (b) , (c) , and (d) .
The lowest energy and shortest conduction path for is shown. The seven-step path begins with a rotation to a binding site on a perpendicular plane followed by a rotation back to the main plane in the B face . Interoctahedral transfer in the B face is followed by two rotations moving the proton to a perpendicular face and back to the plane in the C’ face. Finally two intraoctahedral transfers complete the path. This path is denoted by .
The ten-step path shown below was found in a previous paper (Ref. 3). As in Fig. 4, all rotation arrows involve a rotation to a binding site on a perpendicular plane followed by a rotation back to the main plane. Rotations over and under the plane are dotted and dashed, respectively. This path is summarized by . The thick square indicates the area included in our adjacency matrix. This path is periodic in the larger box. It is also higher in energy and longer than the path shown in Fig. 4.
The spectra of (dashed) is redshifted relative to the spectra (solid). The spectra were normalized by area for plotting in the same scale.
The lattice constants for and found with DFT/PW91, PBE, and LDA compare well to those found in experiment (Refs. 5 and 6). For , the optimized lattice sizes using PW91 with a cutoff of are 5.80, 5.86, and , in agreement with Shi et al. (Ref. 13). Increasing the cutoff to yields the values in the table. Not much change is seen for cutoffs in excess of .
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